(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Development and Simulation of Mathematical Modelling of Hydraulic Turbine"

ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1 



Development and Simulation of Mathematical 
Modelling of Hydraulic Turbine 

Gagan Singh and D.S. Chauhan 
^ttarakhand Technical University, Dehradun, (U.K), India. 

hod_ee@dit.edu.in 
2 Uttarakhand Technical University, Dehradun, (U.K),India. 

pdschauhan @ gmail . com 



Abstract- Power system performance is affected by dynamic 
characteristics of hydraulic governor-turbines during and 
following any disturbance, such as occurrence of a fault, 
loss of a transmission line or a rapid change of load. Accurate 
modelling of hydraulic System is essential to characterize 
and diagnose the system response. In this article the 
mathematical modeling of hydraulic turbine is presented. The 
model is capable to implement the digital systems for 
monitoring and control replacing the conventional control 
systems for power, frequency and voltage. This paper presents 
the possibilities of modeling and simulation of the hydro power 
plants and performs an analysis of different control structures 
and algorithms. 

Key words: mathematical modeling, simulation, hydraulic 
turbine. 

I. INTRODUCTION 

Usually, a typical hydroelectric power plant consists of 
water tunnel, penstock, surge tank, hydraulic turbine, speed 
governor, generator and electric network. Hydrodynamics 
and mechanoelectric dynamics are all involved in such a 
nonlinear dynamic system. Generally, a hydroelectric 
generating unit has many different operating conditions and 
change in any operating condition results in small or large 
hydraulic transients. There are many instances of damage to 
penstock or hydraulic turbine which are most probably 
occurring due to large transients. Hydrodynamics is 
influenced by the performance of hydraulic turbine which 
depends on the characteristics of the water column feeding 
the turbine. These characteristics include water inertia, water 
compressibility and pipe wall elasticity in the penstock. 
Different construction of hydropower systems and different 
operating principles of hydraulic turbines make difficult to 
develop mathematical models for dynamic regime, in order to 
design the automatic control systems. Also, there are major 
differences in the structure of these models. Moreover, there 
are major differences due to the storage capacity of the 
reservoir and the water supply system from the reservoir to 
the turbine (with or without surge chamber). The dynamic 
model of the plants with penstock and surge chamber is more 
complicated than the run-of-the-river plants, since the water 
feed system is a distributed parameters system. This paper 
will present several possibilities for the modeling of the 
hydraulic systems and the design of the control system. 

II. MODELING OF THE HYDRAULIC SYSTEM 

For run-of-the river types of hydropower plants have a 

©2011 ACEEE 
DOL01.UCSI.02.02.1 



low water storage capacity in the reservoir; therefore the 
plant operation requires a permanent balance between the 
water flow through turbines and the river 



i 



4*uflbl -J 



ftffauifi 



I 



*BHw 






Hfr - 



411 

!■] 



Figure 1: Functional block diagram of hydraulic governor-turbine 
system interconnected with a power system network. 

flow in order to maximize the water level in the reservoir for a 
maximum efficiency of water use. Next, we will determine the 
mathematical model for each component of the hydropower 
system. 

A. Mathematical modeling of Hydraulic turbine 

The representation of the hydraulic turbine and water 
column in stability studies is usually based on the following 
assumptions:- 

> The hydraulic resistance is negligible. 

> The penstock pipe is inelastic and the water is 
incompressible. 

> The velocity of the water varies directly with the 
gate opening and with the square root of the net 
head. 

> The turbine output power is proportional to the 
product of head and volume flow. 

The hydraulic turbine can be considered as an element 
without memory since the time constants of the turbine are 
less smaller than the time constants of the reservoir, penstock, 
and surge chamber, if exists, which are series connected 
elements in the system. As parameters describing the mass 
transfer and energy transfer in the turbine we will consider 
the water flow through the turbine Q and the moment M 
generated by the turbine and that is transmitted to the 
electrical generator. These variables can be expressed as non- 
linear functions of the turbine rotational speed N, the turbine 
gate position Z, and the net head H of the hydro system. 
Q = Q(H,N,Z) (1) 

M = M(H,N,Z) (2) 

Through linearization of the equations (1) and (2) around 
the steady state values, we obtain: 



55 



-ArACEEE 



ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1 



&Q = 3 JL &H + *R &N + 3 JL AZ 

^ 3H 3N 3z 

q= ajj A+ fljj w+ fljj 2 

3H 3jY 3Z 

fl2= flj j A+ iJjJ JT+ iJjJ 2 



(3) 



(4) 



Where the following notations were used: 

_ — _ - V _ * M r. _ iJV ** 

^ — — , ?1 — , TYl — , fl — , Z — — 

Qti iV[] Mg Hj^ Z[] 

Which represent the non-dimensional variations of the 
parameters around the steady state values. 

B. The hydraulic feed system 

The hydraulic feed system has a complex geometrical 
configuration, consisting of pipes or canals with different 
shapes and cross-sections. Therefore, the feed system will 
be considered as a pipe with a constant cross-section and 
the length equal with real length of the studied system. In 
order to consider this, it is necessary that the real system and 
the equivalent system to contain the same water mass. Let 
consider m , m 2 .. .m n the water masses in the pipe zones having 
the lengths L,L,...,l and cross- sections A „ A^,...,A of the real 

V 2' ' n V 2' ' n 

feed system. The equivalent system will have the length 
L=l 1 -\-l 2 -\-...+l n and cross-section A, conveniently chosen. 
In this case, the mass conservation law in both systems will 
lead to the equation: 

Since the water can be considered incompressible, the flow 
Q. through each pipe segment with cross-section A. is 
identical and equal with the flow Q through the equivalent 
pipe 

Q=v.A=Qi=Yi.A^ fori=l : 2 : ... : n (6) 

Where v is the water speed in the equivalent pipe, and v. is 
the speed in each segment of the real pipe. 
From the mass conservation law it results: 



v — 1 = fr^L _ £k 

A T.i-i'Ai Y.i-i'Ai 



n 



The dynamic pressure loss can be computed considering the 
inertia force of the water exerted on the cross-section of the 
pipe: 

F; = — m. a = — L. A. p. a = —A — — (S) 

g dt 

Where L is the length of the penstock or the feed canal, A is 
the cross-section of the penstock, g? is the specific gravity 
of water (1000Kgf/m 3 ), a is the water acceleration in the 
equivalent pipe, and g=9.81 m/s 2 is the gravitational 
acceleration. The dynamic pressure loss can be expressed 
as: 



_ F[ _ £-L dv _ y-L Y.[j dQ 



V — Li — LZl _ — 



S I liJ.i dt 



P) 



Using non-dimensional variations, from (9) it results: 



©2011 ACEEE 
DOL01.UCSI.02.02.1 



4tf d _ _pn_ pLElj d {\jJ 
tf dn ~ ** dn ' E kr*i ' dt 
Or in non-dimensional form: 



(10) 



** = -T w * 



■S ^ 

Where T w is the integration constant of the hydropower 
system and the variables have the following meaning: 

* dD QU * dD I IlAL L J X J 

It must be noted that this is a simplified method to compute 
the hydraulic pressure loss, which can be used for run-of-the 
river hydropower plants, with small water head. If an exact 
value of the dynamic pressure is required, then the formulas 
presented in [8], sub-chapter 8.4 "The calculation of hydro 
energy potential" shall be used. 
Using the Laplace transform in relation (1 1), it results: 

h d ($) = -sT w .qlj},tmd <yfc) = - — h d (s') (13) 

Replacing (13) in (3) and (4) and doing some simple 
calculations, we obtain: 






hi'W) = - 



7 



vU) - 






*fc> 



(14) 



(15) 



The mechanical power generated by the turbine can be 
calculated with the relation P-^.g.Q.H, which can be used to 
obtain the linearized relations for variations of these values 
around the steady state values: 

Where fj? is the turbine efficiency, and g, Q, and H were 
defined previously. 

On the other hand, the mechanical power can be determined 
using the relation P=Mu=2?z:M.N, which can be used to obtain 
the linearized relations for variations of these values around 
the steady state values: 



7i = p — m 



(IS) 



56 



Where P D m w is the steady state power generated by 
the turbine for a given steady state flow Q and a steady 
state head H Q , and N is the steady state rotational speed. 
Using these relations, the block diagram of the hydraulic 
turbine, for small variation operation around the steady state 
point, can be determined and is presented in Figure 2, where 
the transfer functions for different modules are given by the 
following relation: 

*- W = i^ T A « ^/"/ ^ = TTTT^< 

14-0. . T. j. J h - 14-CL. . T^.j' 

(-^^-) (19) 

*k ACEEE 



ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1 



For an ideal turbine, without losses, the coefficients a.. 

y 

resulted from the partial derivatives in equations (12 - 16) 
have the following values: a n =0.5; a 12 =a 13 =l; a 21 =1.5; a 23 =l. 
In this case, the transfer functions in the block diagram are 
given by the following relation: 



Table I. Variations of the time constant of the hydro system 



JU*> = : 



-jU*> = : 



-.ff^Cs) = — 



H,J S y = SO—, fl^C) = (l,5 *£-) 



"-w=( i -^y=s 



D.5."T W S 



(20) 



(21) 



(»> 



III. SIMULATION RESULTS 

Example. Let consider a hydroelectric power system with 
the following parameters: 
-Water flow (turbines): Q N =725 m 3 /s; 
-Water level in the reservoir: H N =30 m; 
-The equivalent cross-section of the penstock A=60m 2 ; 
-Nominal power of the turbine P N =178MW; 
-Turbine efficiency rp0.94; 
-Nominal rotational speed of the turbine= 
N=71.43rot/min; 
-The length of the penstock l=SL=20m; 



*Utf 



*M 



JUtf 



HfcW 



— J j — jiki — J 



2 J *Utt| I 



r'r.X, P 




Fig. 2. The block diagram of the hydraulic turbine. 

It shall be determined the variation of the time constant T w 

w 

for the hydro power system. 

For the nominal regime, using relation (12), where Jl.=20m, 

the time constant of the system is: 



r _ "5 



:; 



3D 9..BL.6D 



= 0,82s 



(23) 



Next there is a study of the variation of the time constant 
due to the variation of the water flow through the turbine for 
a constant water level in the reservoir, H=30m, as well as the 
variation due to the variable water level in the reservoir for a 
constant flow Q=725 m 3 /s. In table I, column 3 and figure 3 a) 
are presented the values and the graphical variation of the 
time constant T w for the variation of the water flow between 

w 

500 m 3 /s and 110 m 3 /s, for a constant water level in the 
reservoir, H=30m. In table 1. column 4 and figure 3 b) are 
presented the values and the graphical variation of the time 
constant T w for the variation of the water level in 

w 

©2011 ACEEE 
DOL01.UCSI.02.02.1 



H 


Q 


Tn(H=3Qm) 


Tw«^7:Smc-s 


17 


L 1 *x*5 


I.25fi0fi5 


1.445101 


20 


5NS.14 


1,093155 


1 ,i3 1 736 


23 


&39 r 25 


Q;9:5Q5tf9 


1,071075 


2* 


742.42 


0.S403S3: 


.9WS9 


29 


6G5„e: 


0.753S99S 


0849473 


32 


G03.2I 


0.0832217 


0,709835 


35 


551,51 


0,G24tf598 


0,703849 


J8 


~--r- r 


r 575344C 


0.G4S2S2 



the reservoir, for a constant water flow, Q=725m 3 /s. 




500 6W -00 WO WO 1000 !IW 1*00 

T* [»] 

1,1 1 T 







hi 



W [ni| 40 



Figure 3. Variation of the integral time constant TW: a) by the 
flow Q, b) by the water level H 

It can be seen from the table or from the graphs that the 
time constant changes more than 50% for the entire 
operational range of the water flow through the turbine or if 
the water level in the reservoir varies. These variations will 
create huge problems during the design of the control system 
for the turbine, and robust control algorithms are 
recommended. 

In figure 4 the block diagram of the turbine's power con- 
trol system, is presented using a secondary feedback from 
the rotational speed of the turbine. It can be seen from this 
figure that a dead-zone element was inserted in series with 
the rotational speed sensor in order to eliminate the feedback 
for ±0.5% variation of the rotational speed around the syn- 
chronous value. This oscillation has no significant influence 
on the performance of the system but would have lead to 



57 



^cACEEE 



ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1 



permanent perturbation of the command sent to the turbine 
gate. 



%E1 






j- 1 ' 'LTidE 




Figure 4. Block Diagram of the control system for hydraulic 
turbines 

The constants of the transfer functions had been 
computed for a nominal regime T w =0.8s. The optimal 
parameters for a PI controller are: K R =10, T=0.02s. The results 
of the turbine simulation for different operational regimes are 
presented in figure 5, for a control system using feedbacks 
from the turbine power and rotational speed, with a dead- 
zone on the rotational speed channel for ±0.5% variation of 
the rotational speed around the synchronous value (a) Power 
variation with 10% around nominal value, b) Rotational speed 
variation for power control). 



■M 




Figure 5 Control structure with feedbacks from turbine power and 
rotational speed: 

©2011 ACEEE 
DOL01.UCSI.02.02.1 



In figure 6 the variations of the turbine power (graph a) and 
rotational speed (graph b) for the control system a feedback 
from the turbine power but no feedback from the rotational 
speed are presented. 



(14 




Figure 6 Control structure with only power feedback 

a) Power variation with 10% around nominal value 

b) Rotational speed variation for power control 

IV. CONCLUSIONS 

The detailed mathematical modeling of hydraulic turbine 
is vital to capture essential system dynamic behavior .The 
possibility of implementation of digital systems for monitoring 
and control for power, frequency and voltage in the cascade 
hydro power plant was discussed. The simplified 
mathematical models, capable to accurately describe dynamic 
and stationary behavior of the hydro units a developed and 
simulated. These aspects are compared with experimental 
results. Finally, a practical example was used to illustrate the 
design of controller and to study the system stability. 

Acknowledgement 

The authors would like to thank Professor S.P. Singh (LIT 
Roorkee, India) for his continuous support and valuable 
suggestions. 



58 



4? ACEEE 



ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011 



References 

[1] P. Kundur. Power System Stability and Control. McGraw- 
Hill, 1994. 

[2] IEEE. Hydraulic turbine and turbine control models for system 
dynamic studies. IEEE Transactions on Power Systems, 7(1): 167- 
179, Feb 1992. 

[3] Jiang, J. Design an optimal robust governor for hydraulic turbine 
generating units IEEE Transaction on EC 1, Vol.10, 1995, pp. 18 8- 
194. 

[4] IEEE Working Group. Hydraulic turbine and turbine control 
models for system dynamic studies. IEEE Transactions on Power 
Syst 1992; 7:167-79. 

[5] Nand Kishor, R.P. Saini, S.P. Singh A review on hydropower 
plant models and control, Renewable and Sustainable Energy 
Reviews 11 (2007)776-796 



[6] Asal H. P., R. Widmer, H. Weber, E. Welfonder, W. Sattinger. 

Simulation of the restoration process after black Out in the Swiss 

grid. Bulletin SEV/VSE 83, 22, 1992, pp. 27-34. 

[7] Weber, H., F. PriUwitz, M. Hladky, H. P. Asal. Reality oriented 

simulation models of power plants for restoration studies. Control 

Engineering practice, 9, 2001, pp. 805-811. 

[8] Weber, H., V Fustik, F. Prillwitz, A. Iliev. Practically oriented 

simulation model for the Hydro Power Plant "Vrutok" in Macedonia. 

2nd Balkan Power Conference, 19.-21.06. 2002, Belgrade, 

Yugoslavia 

[9] C Henderson, Yue Yang Power Station - The Implementation of 

the Distributed Control System, GEC Alsthom Technical Review, 

Nr. 10, 1992. 

[10] Prillwitz E, A. Hoist, H. W. Weber (2004), "Reality Oriented 

Simulation Models of the Hydropower Plants in Macedonia and 

Serbia/Montenegro" In Proceedings of the Annual Scientific Session 

TU Verna, Bulgaria. 



©2011 ACEEE 
DOL01.UCSI.02.02.1 



59 



^VACEEE